论文标题
分支组的亚组感应属性
Subgroup induction property for branch groups
论文作者
论文摘要
亚组感应特性是作用于Grigorchuk和Wilson在2003年引入的生根树上的自相似群体的属性，该群体似乎对拥有它的群体的结构具有很强的影响。例如，在证明中使用的是第一个Grigorchuk组以及Gupta-Sidki 3组是可分离的（局部扩展的残留有限），或描述其有限生成的亚组以及它们弱最大的亚组。但是，到目前为止，只有两个已知的与该特性的群体的例子，即第一个Grigorchuk组和Gupta-Sidki 3组。 本文的目的是双重的。首先，我们研究了分支组的亚组感应特性的各种后果，分支组是一类特别有趣的自相似群体。值得注意的是，我们表明，具有亚组感应特性的有限生成的分支组必须是扭转，只有无限和亚组可分离，并且我们建立了所有最大亚组均为有限指数，并且它们所有弱最大的亚组在prodinite拓扑中都封闭。然后，我们表明，每个扭力GGS组都有亚组感应属性，因此提供了第一个无限的群体示例家族。
The subgroup induction property is a property of self-similar groups acting on rooted trees introduced by Grigorchuk and Wilson in 2003 that appears to have strong implications on the structure of the groups possessing it. It was for example used in the proof that the first Grigorchuk group as well as the Gupta-Sidki 3-group are subgroup separable (locally extended residually finite) or to describe their finitely generated subgroups as well as their weakly maximal subgroups. However, until now, there were only two known examples of groups with this property, namely the first Grigorchuk group and the Gupta-Sidki 3-group. The aim of this article is twofold. First, we investigate various consequences of the subgroup induction property for branch groups, a particularly interesting class of self-similar groups. Notably, we show that finitely generated branch groups with the subgroup induction property must be torsion, just infinite and subgroup separable, and we establish conditions under which all their maximal subgroups are of finite index and all their weakly maximal subgroups are closed in the profinite topology. Then, we show that every torsion GGS group has the subgroup induction property, hence providing the first infinite family of examples of groups with this property.
